A classification of rational 3-tangles
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918034234408960 |
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| author | Kwon, Bo-hyun |
| author_facet | Kwon, Bo-hyun |
| contents | In this paper, we define the \textit{normal form} and \textit{normal coordinate} of a rational 3-tangle $T$ with respect to $\partial E_1$, where $E_1$ is the fixed two punctured disk in $Σ_{0,6}$. Among all normal coordinates of $T$ with respect to $\partial E_1$, we investigate the collection of \textit{minimal} normal coordinates of $T$. We show that the simplicial complex constructed with normal forms of the rational 3-tangle is contractible. As an effectiveness of the contractibility of the simplicial complex by normal forms of $T$, we would choose a minimal normal coordinate of $T$ with a certain rule for the representative for the rational $3$-tangle $T$. This classifies rational $3$-tangles up to isotopy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_19082 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A classification of rational 3-tangles Kwon, Bo-hyun Geometric Topology In this paper, we define the \textit{normal form} and \textit{normal coordinate} of a rational 3-tangle $T$ with respect to $\partial E_1$, where $E_1$ is the fixed two punctured disk in $Σ_{0,6}$. Among all normal coordinates of $T$ with respect to $\partial E_1$, we investigate the collection of \textit{minimal} normal coordinates of $T$. We show that the simplicial complex constructed with normal forms of the rational 3-tangle is contractible. As an effectiveness of the contractibility of the simplicial complex by normal forms of $T$, we would choose a minimal normal coordinate of $T$ with a certain rule for the representative for the rational $3$-tangle $T$. This classifies rational $3$-tangles up to isotopy. |
| title | A classification of rational 3-tangles |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2505.19082 |